Follow a result back to its evidence.

SEAM.BIT.ORIGIN.01: the tautology attack on the v506 alignment bit, executed and refuted -- where do the marks AND the deck come from, and does their common origin force the central alignment? (Part A torus origin + Part B fermion dichotomy in one module; NO marker moves -- the bit stays [C], SEAM.CLOCK.RIGIDITY.01 and QGEO.REALIZE.01 text-precised only.) Exact sympy + exact Cl(16) Fractions, no floats. [E] ORIGIN THEOREM: the 4 marks are the fixed points of the hyperelliptic involution w -> -w on C/(Z + tau Z) = the four half-periods Lambda/2Lambda (a 1/3-coefficient never qualifies); a translation t_c descends to the base sphere iff 2c in Lambda -- exactly the THREE half-period translations (control c = 1/3 fails); the descended maps are the EXPLICIT rational involutions sigma_i(x) = e_i + (e_i-e_j)(e_i-e_k)/(x-e_i) with the exact curve-lift identity prod(sigma_i - e_l) = psi_i^2 prod(x - e_l), psi_i RATIONAL (the p-function addition formula as pure algebra); {id, sigma_1, sigma_2, sigma_3} = Klein V4; CONVERSELY, in ALL three stabiliser types the freely-acting involutions are EXACTLY the three sigma_c (generic V4 (4,3,3), harmonic D4 (8,5,3), hexagonal A4 (12,3,3)) => EVERY mark-free collar deck (v216) IS a half-period translation -- the CLASS of the deck is forced, its POSITION is not. [E] CORE SOLVE (symbolic in the Legendre modulus lambda): sigma_c admits a marks-preserving PGL2 square root <=> lambda in {-1} (sigma_1), {2} (sigma_2), {1/2} (sigma_3) -- per half-period exactly ONE solution; intrinsically: the two deck pairs separate each other HARMONICALLY (pair cross-ratio = -1, same solution sets); the union over c is the harmonic orbit {-1, 2, 1/2} <=> j = 1728 <=> tau = i; generic modulus: Stab = V4 abelian, ALL sigma_c central and yet clockless ('central alone is EMPTY' -- the bit is the CONJUNCTION Stab = D4 AND deck central); at tau = i the stabiliser jumps to D4 with sigma_1 central (2 roots = the clock pair) and sigma_2/sigma_3 edge-type (0 roots); the v506 seam deck z -> -z lands on the central sigma_1 = -1/x under ALL 8 mark bijections; CM DICTIONARY: mult-by-i fixes exactly ONE nonzero half-period c* = (1+tau)/2 (mult-by-rho 3-cycles all three -- hexagonal has no distinguished one); the clock lift needs the CM unit i (psi = 2i/(x+1)^2, t^2+1 irreducible over Q) while half-period decks lift RATIONALLY: the clock root exists only WITH CM by Z[i] <=> tau = i. [E] EQUIVALENCE TETRAD + CIRCULARITY AUDIT: 'clock exists' <=> (tau = i AND deck = the CM-fixed half-period translation) <=> (Stab = D4 AND deck central) <=> deck pairs harmonically separated <=> the NS Fock lift is NONSPLIT (Part B); audit: the v493 converse (clock GIVEN => P(iZ) = P(Z) forces a3 = a2 = a1 = 0, j = 1728) independently re-verified -- two independently computed converses of ONE equivalence, no circularity. [E] FERMION DICHOTOMY (the new physical face): central arrangement (shift by 8 on the 16-site NS seam): Utilde^2 = 256 gamma_1...gamma_16 => U^2 = (-1)^F, nonsplit Z4 extension, 2 square roots in the 32-element dihedral lift group (the Z8 tower); edge arrangement (reflection j -> 4-j, fixed sites = NON-marks = the silver {0, inf} between neighbour marks): the implementation EXISTS but Utilde_e^2 = +2^7 x 1 => U^2 = +1, the extension SPLITS, 0 roots, no Z8 tower; one-particle level (lam^2+1)^8 vs (lam^2-1)^8; arrangement census: ALL 28 shift-invariant 4-sets cross (central type), ALL 21 mark-free reflection-invariant sets do not; robustness at 8 sites; R-sector edge control splits too -- the bit IS a physical extension class (Fidkowski-Kitaev-type): distinguished, but not derived; WHICH class nature realises stays the input. [E] SILVER PLACEMENT (in-family witness against the tautology): the v506 counterexample {+-1, +-(3+2sqrt2)} = 'tau = i with deck = a NON-CM-fixed half-period translation' (all 8 mark bijections carry its deck to sigma_2/sigma_3, never sigma_1) -- the common torus origin does NOT exclude the misalignment. [C] THE REST (verbatim, no overclaim): the one alignment bit remains a carrier datum; no tautology -- the origin forces the CLASS of the deck (half-period translation), not its POSITION. Exact (sympy + Cl(16) Fractions).

Linked claims

SEAM.BIT.ORIGIN.01

Paper sections

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